Representation of numbers with a single 4

Representing numbers as a result of arithmetic operations on a restricted set of numbers is an entertaining activity (Three 3's, Three 4's, Three 5's, Four 3's, Four 4's, Four 5's) that even lends itself to some degree of systematization. Since the basic arithmetic operations take two arguments (a+b, a*b, ab, etc.), it never occurred to me to consider representing numbers with a single selected number. For instance, 1 = [3], 2 = [3!], 3 = 3. However, I received the following letter from one of the visitors:

I have been interested in this puzzle for thirty years. Is it possible to represent the numbers 1 - 12 mathematically only using 1 four?

  1. 4
  2. F4 (Fibonacci number 4)
  3. 4
  4. B4 (Bell number 4)
  5. g4 (gnomic number 4)
  6. 4!! (double factorial 4)
  7. !4 (subfactorial 4)
  8. T4 (triangular number 4)(*)
  9. 4$ (superfactorial 4). It was noted by Craig Katz that the superfactorial of 4, as defined below is rather 288. In fact, 3$ = 12.

Regards,

Robert Smith

Well, there are at least two ways to look at the table. For one, I'll use it as an index to introduce the number families referred to by Robert (Fibonacci numbers, Bell numbers, ...). Secondly, since accepting notations that index such families of numbers (e.g., F4) would mean going beyond the original problem of representing numbers with arithmetic operations, I consider this a challenge to put up a similar table while staying in the framework of arithmetic operations.

Below are the results of my first attempt. You are welcome to suggest more entries. I would still allow for supefactorial and subfactorial notations, although they are far from being standard. To shorten the representations, I shall use the conventions introduced for the same purpose elsewhere. Thus {n} refers to an entry in row #n provided, of course, it has already been filled. (exp(x) = ex and its inverse ln(x) are being used somewhat reluctantly.)

1 [4]
2 4
3 !4
4 4
5 [[(4!)!]]
6 {3}!
7 [{10}!!], [e4](2)
8 4!!
9 !4
10 [{5}!]
11 [!{8}](1)
12 [ln({9}!)]
13 [ln((4!)!)](2)
14 [ln !{10}](3)
15 5!!(2)

(1) By Don Gosiewski

(2) By Andre Gustavo dos Santos

(3) By Don

Double Factorial

n! is the product of all integers less or equal to n that have the same parity as n. 7!! = 7*5*3*1, 10!! = 10*8*6*4*2.

Superfactorial

n$ is the product of all factorials k! of integers not exceeding n:

Subfactorial

!n is the number of Derangements (i.e., permutations in which no number appears in its original place.) There are several ways to define this number. One relates it to the factorial:

The number was discovered independently by Niclaus Bernoulli and L.Euler. The idea of derangements is exemplified in a concrete form by the following problem of misaddressed letters:

Someone writes n letters and prepares n envelopes with the corresponding addresses. How many ways are there of placing all the letters into the wrong envelopes?

Bell Numbers

Bell numbers Bn are named after Eric Temple Bell, a mathematician and a prolific mathematical historian. Bn is the number of possible arrangements of n distinct objects into groups.

Robert Smith is mistaken about B4 which in fact is 15. Indeed, there's 1 group that combines all four distinct objects, 4 different arrangements into two groups of one and three objects, 3 different arrangements into two groups of two objects each, 6 arrangements into three groups of one, one and two objects, and 1 arrangement into four groups each containing a single object.

Gnomic Numbers

Ancient Greeks defined gnomon as that figure which, when added to another figure, produces the figure similar to the original. For example, n2 + (2n+1) = (n+1)2. We may therefore call odd numbers square gnomons.

(*)

Originally Robert Smith referred to T4 as a tetrahedral number. Vladimir Zajic has pointed out that T4 is the standard symbol for the fourth triangular number 4·(4 + 1)/2, which incidently is equal to 10. The usual notation for the tetrahedral numbers is Ten or Tetn (not Tn) and they are defined as Ten = Snk=1 Tk. The fourth tetrahedral number equals 1 + 3 + 6 + 10 = 20.

Vladimir also notes that the fourth Lucas number L4 = 7. Lucas numbers are defined by the same recurrence formula as Fibonacci numbers: Ln+1 = Ln + Ln-1 but with a different seed (L1 = 1, L2 = 3) as opposed to the Fibonacci sequence seed (F1 = 1, F2 = 1).


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